4 BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS 335
(15.4.7) Let E be a complex Hilbert space. Then the Banach algebra
A = «£?(£) (15.1.6) is a star algebra with respect to the involution wh-»w*
(11.5).
Hilbert-Schmidt Operators
(15.4.8) Let E be an infinite-dimensional separable complex Hilbert space.
An operator u e £? (E) is said to be a Hilbert-Schmidt operator if, for one
choice of a Hilbert basis (#„), the series with general term |lw(<2n)||2 is con-
vergent. If (&„) is another Hilbert basis of E, then ||w(an)||2 = £ \(u(an) I *m)l2
m
by Parseval's identity (6.5.2), and therefore the series £ l|w(an)||2 converges
n
if and only if the double family (\(u(an)\bm)\2) *s summable (5.3.4).
by Parseval's identity. Hence an equivalent condition is that the series with
general term || u*(bn)\\2 should be convergent. It follows that, if w is a Hilbert-
Schmidt operator, the series with general term ||w(^)||2 converges for all
choices of the Hilbert basis (#„), that w* is a Hilbert-Schmidt operator, and
that the number
(15.4.8.1) Nl2 = (III«(a«)lrl =II"*II2
is independent of the Hilbert basis (an) chosen. We shall show that the set
J*?2(E) of Hilbert-Schmidt operators is an algebra, on which \\u\\2 is a norm,
and that, endowed with this norm and the involution u\—»u*, the algebra
j*?2(E) is a Banach algebra with involution, without unit element.
The fact that w is a Hilbert-Schmidt operator also means that, if (an) is
any Hilbert basis of E, the sequence (||w(#n)||) eRN belongs to the Hilbert
space /R (6.5), and the norm of this sequence in /£ is equal to ||w||2. If
u, i?eJ$?2(E), then the inequality \\(u + v)(an)\\ <jj \\u(an)\\ + \\v(an)\\ and the
preceding remark show that u H- v e J^2(E) and that \\u + v\\2 ^ \\u\\2 + ||u||2.
Clearly also Aw e &2(E) for all A e C, and ||Aw||2 = |A| • \\tt\\2.
For every continuous operator w e J£(E), the operator w o u is a Hilbert-
Schmidt operator, by reason of the inequality
l|H<w(0B))|| <£ ||w|| • \\u(an)\\
(5.5.1), and we have
(15.4.8.2) l|woW||2g ||w|| • ||w||2.1,